*The distance from \(x\) to 5 can be represented using the absolute value as \(|x−5|\).We want the values of \(x\) that satisfy the condition \(| x−5 |\leq4\).Instead, the width is equal to 1 times the vertical distance as shown in Figure \(\Page Index\).*

This means that the corner point is located at \((3,4)\) for this transformed function.

Solution The basic absolute value function changes direction at the origin, so this graph has been shifted to the right 3 units and down 2 units from the basic toolkit function. We also notice that the graph appears vertically stretched, because the width of the final graph on a horizontal line is not equal to 2 times the vertical distance from the corner to this line, as it would be for an unstretched absolute value function.

Let's work through some examples to see how this is done. Another benefit of this graphing technique is that you do not need to verify any of the solutions--since we are only graphing the pieces that are actually mathematically possible, we get all the solutions we are looking for, no less and no more.

If you could not discern the solutions from the picture, you can simply solve the equation for each case.

In this case, we can see that the isolated absolute value is to be less than or equal to a negative number.

Again, the absolute value will always be positive; hence, we can conclude that there is no solution.Solution Using the variable \(p\) for passing, \(| p−80 |\leq20\) Figure 1.6.4 shows the graph of \(y=2|x–3| 4\).The graph of \(y=|x|\) has been shifted right 3 units, vertically stretched by a factor of 2, and shifted up 4 units.Until the 1920s, the so-called spiral nebulae were believed to be clouds of dust and gas in our own galaxy, some tens of thousands of light years away.Then, astronomer Edwin Hubble proved that these objects are galaxies in their own right, at distances of millions of light years.Solution We want the distance between \(x\) and 5 to be less than or equal to 4.We can draw a number line, such as the one in , to represent the condition to be satisfied.Today, astronomers can detect galaxies that are billions of light years away.Distances in the universe can be measured in all directions.A very basic example would be as follows: if required.However, these problems are often simplified with a more sophisticated approach like being able to eliminate some of the cases, or graphing the functions.

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